Metric projections and the differentiability of distance functions

Fitzpatrick, Simon

doi:10.1017/s0004972700006596

Cited by 73 publications

(43 citation statements)

References 14 publications

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“…If, in addition, the norm of E* is Fréchet differentiable at u* then the corresponding metric projection Pm is continuous at x. This uses and improves on results from [2].…”

Section: Introductionmentioning

confidence: 72%

“…We call a sequence (yn) from M a minimizing sequence for x provided \\x -yn\\ -y ¡pm(x) as n -» oo and we say that Pm is continuous at x provided yn -> yo whenever yn G Pm(xti) for all n > 0 and xn -> xq. If every minimizing sequence for x converges then Pm is continuous at x; the converse holds in Banach spaces whose norms are sufficiently well behaved (see [2]). …”

Section: Introductionmentioning

confidence: 99%

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Differentiation of real-valued functions and continuity of metric projections

Fitzpatrick¹

1984

Proc. Amer. Math. Soc.

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ABSTRACT. We characterize the Freenet differentiability of real-valued functions on certain real Banach spaces in terms of a directional derivative being equal to a modified version of the local Lipschitz constant.This yields the continuity of metric projections onto closed sets whose distance functions have directional derivatives equal to 1, provided the Banach space and its dual have Fréchet differentiable norms.

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“…If, in addition, the norm of E* is Fréchet differentiable at u* then the corresponding metric projection Pm is continuous at x. This uses and improves on results from [2].…”

Section: Introductionmentioning

confidence: 72%

Section: Introductionmentioning

confidence: 99%

Differentiation of real-valued functions and continuity of metric projections

Fitzpatrick¹

1984

Proc. Amer. Math. Soc.

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“…It is known that P S (u) is a singleton, whenever the distance function d S (·) is Fréchet differentiable at u ∈ cl S, see also [13,Theorem 2.4], a part of the proof of the Remark below can be also used to get it. We use the idea behind the result to get a local sequential weak closedness of S, that is we prove that (11) holds true, whenever d S (·) is Fréchet differentiable at u ∈ cl S. Remark 3.4 Let H be a real Hilbert space, S ⊂ H be a nonempty subset, u ∈ S, S ⊂ cl S be given ands ∈ S .…”

Section: Remark 33mentioning

confidence: 99%

“…Another way to get the convexity of a Chebyshev set is to assume a differentiability of the distance function outside the set. Namely, if the distance function to a Chebyshev set is Fréchet differentiable at all points outside the set then the set is convex too, we refer to [13,14,18,19] for details. Due to L. P. Vlasov we know also that the continuity of the metric projection (which implies the convexity) can be obtained by checking if the Vlasov condition is satisfied, see (22) and [35, page 56], we refer also to [33,34].…”

Section: Some Sufficient Conditions For the Convexity Of Chebyshev Setsmentioning

confidence: 99%

On Closures of Preimages of Metric Projection Mappings in Hilbert Spaces

Zagrodny

2015

Set-Valued Var. Anal

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The closure of preimages (inverse images) of metric projection mappings to a given set in a Hilbert space are investigated. In particular, some properties of fibers over singletons (level sets or preimages of singletons) of the metric projection are provided. One of them, a sufficient condition for the convergence of minimizing sequence for a giving point, ensures the convergence of a subsequence of minimizing points, thus the limit of the subsequence belongs to the image of the metric projection. Several examples preserving this sufficient condition are provided. It is also shown that the set of points for which the sufficient condition can be applied is dense in the boundary of the preimage of each set from a large class of subsets of the Hilbert space. As an application of obtained properties of preimages we show that if the complement of a nonconvex set is a countable union of preimages of convex closed sets then there is a point such that the value of the metric projection mapping is not a singleton. It is also shown that the Klee result, stating that only convex closed sets can be weakly closed Chebyshev sets, can be obtained for locally weakly closed sets. This paper is dedicated to Professor Lionel Thibault to honor great man and great mathematician. Thank You Lionel for the journeys in mathematics which we have gone together. Keywords

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“…Observe that since the bidual of a space with the Mazur Intersection Property has the w*-Mazur Intersection Property , Theorem 2.1 (b) shows that every u;*-compact convex set in l°°, with the bidual of the above norm, is the w*-closed convex hull of its farthest points. Thus, there is a closed bounded convex set K C Co, such that no farthest point of the iu*-closure of K in X" comes from K. [5] Farthest points 429…”

Section: A>0mentioning

confidence: 99%

Farthest points and the farthest distance map

Bandyopadhyay¹,

Dutta²

2005

Bull. Austral. Math. Soc.

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In this paper, we consider farthest points and the farthest distance map of a closed bounded set in a Banach space. We show, inter alia, that a strictly convex Banach space has the Mazur intersection property for weakly compact sets if and only if every such set is the closed convex hull of its farthest points, and recapture a classical result of Lau in a broader set-up. We obtain an expression for the subdifferential of the farthest distance map in the spirit of Preiss' Theorem which in turn extends a result of Westphal and Schwartz, showing that the subdifferential of the farthest distance map is the unique maximal monotone extension of a densely defined monotone operator involving the duality map and the farthest point map.

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Metric projections and the differentiability of distance functions

Abstract: Examples are presented to show that some of our hypotheses are needed.

Cited by 73 publications

References 14 publications

Differentiation of real-valued functions and continuity of metric projections

Differentiation of real-valued functions and continuity of metric projections

On Closures of Preimages of Metric Projection Mappings in Hilbert Spaces

Farthest points and the farthest distance map

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